Dynamic failure in the extensive plastic-strain range

Abstract

Here we consider failure in thin-walled cylindrical shells made of a material showing not only viscosity but also work hardening. A differential equation is derived with fuller incorporation of expanding-shell motion. The breakup of a jet* of continuous material having a velocity gradient (another example of failure in the deep plastic region) can also be described by tlhe differential equation and solution derived for a shell. Formulation and Solution. Consider the motion of a thin ring of unit width having radius r 0 and thickness 6 o conceptually cut from the cylindrical shell. An observer moving radially from the center with a particle of the material with speed v sees an adjacent particle on another radius at a distance@ r moving with a velocity proportional to the distance. In :fact, in dt the adjacent particle moves a distance (r + vdt)@ with velocity @v. Therefore, :from the observer's viewpoint there is a velocity gradient v/r in the tangential direction. We consider simultaneously a freely moving jet having a constant positive velocity gradient u/m in the direction of motion (u is the velocity difference between the head and tail of the jet and m is the length). The gradients v/r and u/m cause the ring and jet to stretch and thin out. To simplify the discussion, we assume that v and u are constant. This assumption is equivalent to the energy dissipated in plastic flow being small by comparison with the total kinetic energy of the ring or the kinetic energy of the jet material relative to its center of mass.# For the~jet, the speed of the center of mass is independent of u, while in the ring the energy dissipation in plastic flow retards the basic radial motion. This ,difference i:n the energy sources in plastic-flow dissipation must be borne in mind in determining the limitations of the final solution. The velocity gradient in the ring is perpendicular to the radial motion, while that in the jet is parallel to it, but this has no essential importance. The main kinematic difference is that all the phases of motion and failure in the sectors are synchronous for the ring, while they are sequential for the jet, beginning with the head parts. However, even this difference can be avoided with certain simplifying assumptions. As the ring (cylindrical shell) [6] and jet split up into a large number n of fragments (with the failure independent), without loss of generality it is sufficient to consider the ewolution of a ring sector or part of the jet of size Zi/n of the whole on the basis that all the processes are synchronous in such a part and the diameter is dependent only on *The first such attempt was made by S. V. Serikov at the Third All-Union Seminar on Explosion Physics (June 25-29, 1984, Krasnoyarsk). '~The assumption made in [2] for a ring is not too strong also for a jet such as a cumulative one if one bears in mind the reduction in the yield point of the material due to the initial heating to several hundred degrees, since the material undergoes shock compression as the jet is formed and then isentropic unloading.